Lovász [1]} established that \(\mathrm {wsat}(n, K_r)=\binom{n}{2}-\binom{n-r+2}{2}\)  when \(n\geqslant r\geqslant 2\) , settling a conjecture of Bollobás [2]}. The result is also proved by Frankl [3]}, Kalai [4]}, Alon [5]}, and Yu [6]}.
Surprisingly, these proofs all are based on algebraic techniques and no combinatorial proof has been found so far.
